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Register a new userModified Riemann sums of Riemann-Stieltjes integrable functions
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Authors
Alberto Torchinsky
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In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.
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In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums md0 R_nf(x)=frac{1}{n}sum_{k=0}^{n-1}fbigg(x+frac{k}{n}bigg),quad xin ZT. emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $fin L^p$ with $p>1$.
We consider general formulations of the change of variable formula for the Riemann-Stieltjes integral, including the case when the substitution is not invertible.
We consider two integrals over $xin [0,1]$ involving products of the function $zeta_1(a,x)equiv zeta(a,x)-x^{-a}$, where $zeta(a,x)$ is the Hurwitz zeta function, given by $$int_0^1zeta_1(a,x)zeta_1(b,x),dxquadmbox{and}quad int_0^1zeta_1(a,x)zeta_1(b,1-x),dx$$ when $Re (a,b)>1$. These integrals have been investigated recently in cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $zeta(x)$ and two free parameters $a$ and $b$. The limiting forms of these sums when $a+b$ takes on integer values are considered.
We generalize Warnaars elliptic extension of a Macdonald multiparameter summation formula to Riemann surfaces of arbitrary genus.
Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a semi-rational polyhedron and a real sampling parameter t, if we allow for step-polynomial coefficients, instead of just constant ones.
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